Triads in The Chromatic Scale
The major triad can be defined by a pair of intervals from the root note: a major third (interval spanning 4 semitones) and a perfect fifth (7 semitones). The minor triad can likewise be defined by a minor third (3 semitones) and a perfect fifth (7 semitones).
As shown above, a chromatic scale has twelve intervals spanning seven semitones. Eleven of these are perfect fifths (P5), while the twelfth is a diminished sixth (d6). Since they span the same number of semitones, P5 and d6 are considered to be enharmonically equivalent. In an equally tuned chromatic scale, P5 and d6 have exactly the same size. The same is true for all the enharmonically equivalent intervals spanning 4 semitones (M3 and d4), or 3 semitones (m3 and A2). However, in the meantone temperament this is not true. In this tuning system, enharmonically equivalent intervals may have different sizes, and some intervals may markedly deviate from their justly tuned ideal ratios. As explained in the previous section, if the deviation is too large, then the given interval is not usable, either by itself or in a chord.
The following table focuses only on the above-mentioned three interval types, used to form major and minor triads. Each row shows three intervals of different types but which have the same root note. Each interval is specified by a pair of notes. To the right of each interval is listed the formula for the interval ratio. The intervals d4, d6 and A2 may be regarded as wolf intervals, and have been marked in red. S and X denote the ratio of the two above-mentioned kinds of semitones (m2 and A1).
| 3 semitones (m3 or A2) |
4 semitones (M3 or d4) |
7 semitones (P5 or d6) |
|||
|---|---|---|---|---|---|
| Interval | Ratio | Interval | Ratio | Interval | Ratio |
| C—E♭ | S2 · X | C—E | S2 · X2 | C—G | S4 · X3 |
| C♯—E | S2 · X | C♯—F | S3 · X | C♯—G♯ | S4 · X3 |
| D—F | S2 · X | D—F♯ | S2 · X2 | D—A | S4 · X3 |
| E♭—F♯ | S · X2 | E♭—G | S2 · X2 | E♭—B♭ | S4 · X3 |
| E—G | S2 · X | E—G♯ | S2 · X2 | E—B | S4 · X3 |
| F—G♯ | S · X2 | F—A | S2 · X2 | F—C | S4 · X3 |
| F♯—A | S2 · X | F♯—B♭ | S3 · X | F♯—C♯ | S4 · X3 |
| G—B♭ | S2 · X | G—B | S2 · X2 | G—D | S4 · X3 |
| G♯—B | S2 · X | G♯—C | S3 · X | G♯—E♭ | S5 · X2 |
| A—C | S2 · X | A—C♯ | S2 · X2 | A—E | S4 · X3 |
| B♭—C♯ | S · X2 | B♭—D | S2 · X2 | B♭—F | S4 · X3 |
| B—D | S2 · X | B—E♭ | S3 · X | B—F♯ | S4 · X3 |
First, look at the last two columns on the right. All the 7-semitone intervals except one have a ratio of
which deviates by −5.4 cents from the just 3:2 of 702.0 cents. Five cents is small and acceptable. On the other hand, the d6 from G♯ to E♭ has a ratio of
which deviates by +35.7 cents from the just fifth. Thirty-five cents is beyond the acceptable range.
Now look at the two columns in the middle. Eight of the twelve 4-semitone intervals have a ratio of
which is exactly a just 5:4. On the other hand, the four d4 with roots at C♯, F♯, G♯ and B have a ratio of
which deviates by +41.1 cents from the just M3. Again, this sounds badly out of tune.
Major triads are formed out of both major thirds and perfect fifths. If either of the two intervals is substituted by a wolf interval (d6 instead of P5, or d4 instead of M3), then the triad is not acceptable. Therefore major triads with root notes of C♯, F♯, G♯ and B are not used in meantone scales whose fundamental note is C.
Now look at the first two columns on the left. Nine of the twelve 3-semitone intervals have a ratio of
which deviates by −5.4 cents from the just 6:5 of 315.6 cents. Five cents is acceptable. On the other hand, the three augmented seconds whose roots are E♭, F and B♭ have a ratio of
which deviates by −46.4 cents from the just minor third. It is a close match, however, for the 7:6 septimal minor third of 266.9 cents, deviating by +2.3 cents. These augmented seconds, though sufficiently consonant by themselves, will sound "exotic" or atypical when played together with a perfect fifth.
Minor triads are formed out of both minor thirds and fifths. If either of the two intervals are substituted by an enharmonically equivalent interval (d6 instead of P5, or A2 instead of m3), then the triad will not sound good. Therefore minor triads with root notes of E♭, F, G♯ and B♭ are not used in the meantone scale defined above.
The following major triads are usable: C, D, E♭, E, F, G, A, B♭.
The following minor triads are usable: C, C♯, D, E, F♯, G, A, B.
The following root notes are useful for both major and minor triads: C, D, E, G and A. Notice that these five pitches form a major pentatonic scale.
The following root notes are useful only for major triads: E♭, F, B♭.
The following root notes are useful only for minor triads: C♯, F♯, B.
The following root note is useful for neither major nor minor triad: G♯.
Read more about this topic: Quarter-comma Meantone
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